Review: Leaning on the works of H. Poincaré, H. Weyl, A. Heyting, V. F. Asmus and other, this article demonstrates the correlation between the ideas of intuitivism of H. Poincaré and the ideas of intuitionism of L. E. J. Brower and H. Weyl. The subject of this research is the role of intuition in formation of the fundamentals of mathematical science. The author examines the state of mathematical science at the turn of the XIX-XX centuries, as well as the crises of the foundations of mathematics. The firs trend of the philosophy of mathematics pursuing to overcome this crisis became the Russell’s logicism – an attempt to form mathematics on the logical basis, excluding the intuitive elements. Leveling criticism at this trend, H. Poincaré develops a doctrine on the priority of intellectual intuition in the formation of mathematics. The article reveals the heuristic value of Russell’s ideas for both, the advancement of science, and comprehension of the nature of scientific creativity. The author describes their correlation with the ideas of H. Poincaré regarding the priority of intellectual intuition. The ideas of intuitionists, in turn, expressing the moods of scholars with humanistic reasoning, became a fruitful ground for the development of ontological and gnoseological foundations of mathematics.

Keywords: intuitionism, logicism, intuitivism, set theory, Russell's paradox, foundations of mathematics, gnosiology, logic, intuition, becoming object